P. Démoulin

FORMATION OF TORUS-UNSTABLE FLUX ROPES AND ELECTRIC CURRENTS IN ERUPTING SIGMOIDS

We analyze the physical mechanisms that form a three-dimensional coronal flux rope and later cause its eruption. This is achieved by a zero-β magnetohydrodynamic (MHD) simulation of an initially potential, asymmetric bipolar field, which evolves by means of simultaneous slow magnetic field diffusion and sub-Alfvenic, line-tied shearing motions in the photosphere. As in similar models, flux-cancellation-driven photospheric reconnection in a bald-patch (BP) separatrix transforms the sheared arcades into a slowly rising and stable flux rope. A bifurcation from a BP to a quasi-separatrix layer (QSL) topology occurs later on in the evolution, while the flux rope keeps growing and slowly rising, now due to shear-driven coronal slip-running reconnection, which is of tether-cutting type and takes place in the QSL. As the flux rope reaches the altitude at which the decay index –∂ln B/∂ln z of the potential field exceeds ~3/2, it rapidly accelerates upward, while the overlying arcade eventually develops an inverse tear-drop shape, as observed in coronal mass ejections (CMEs). This transition to eruption is in accordance with the onset criterion of the torus instability. Thus, we find that photospheric flux-cancellation and tether-cutting coronal reconnection do not trigger CMEs in bipolar magnetic fields, but are key pre-eruptive mechanisms for flux ropes to build up and to rise to the critical height above the photosphere at which the torus instability causes the eruption. In order to interpret recent Hinode X-Ray Telescope observations of an erupting sigmoid, we produce simplified synthetic soft X-ray images from the distribution of the electric currents in the simulation. We find that a bright sigmoidal envelope is formed by pairs of -shaped field lines in the pre-eruptive stage. These field lines form through the BP reconnection and merge later on into -shaped loops through the tether-cutting reconnection. During the eruption, the central part of the sigmoid brightens due to the formation of a vertical current layer in the wake of the erupting flux rope. Slip-running reconnection in this layer yields the formation of flare loops. A rapid decrease of currents due to field line expansion, together with the increase of narrow currents in the reconnecting QSL, yields the sigmoid hooks to thin in the early stages of the eruption. Finally, a slightly rotating erupting loop-like feature (ELLF) detaches from the center of the sigmoid. Most of this ELLF is not associated with the erupting flux rope, but with a current shell that develops within expanding field lines above the rope. Only the short, curved end of the ELLF corresponds to a part of the flux rope. We argue that the features found in the simulation are generic for the formation and eruption of soft X-ray sigmoids.

Three‐dimensional magnetic reconnection without null points: 1. Basic theory of magnetic flipping

In two or three dimensions, magnetic reconnection may occur at neutral points and is accompanied by the transport of magnetic field lines across separatrices, the field lines (or flux surfaces in three dimensions) at which the mapping of field lines is discontinuous. Here we show that reconnection may also occur in three dimensions in the absence of neutral points at so-called “quasi-separatrix layers,” where there is a steep gradient in field line linkage. Reconnection is a global property, and so, in order to determine where it can occur, the first step is to enclose the volume being considered by a boundary (such as a spherical surface). Then the mapping of field lines from one part of the boundary to another is determined, and quasi-separatrix layers may be identified as regions where the gradient of the mapping or its inverse is very much larger than normal. The most effective measure of the presence of such layers is the norm of the displacement gradient tensor; their qualitative location is robust and insensitive to the particular surface that is chosen. Reconnection itself occurs when there is a breakdown of ideal MHD and a change of connectivity of plasma elements, where the field line velocity becomes larger than the plasma velocity, so that the field lines slip through the plasma. This breakdown can occur in the quasi-separatrix layers with an electric field component parallel to the magnetic field. In three dimensions the electric field E (and therefore the field line velocity v⊥) depends partly on the imposed values of E (or v⊥) at the boundary and partly on the gradients of the inverse mapping function. We show that the inverse mapping determines the location of the narrow layers where the breakdown of ideal MHD can occur, while the imposed boundary values of v⊥ determine mainly the detailed flow pattern inside the layers. Thus, in general, E (and therefore v⊥) becomes much larger than its boundary values at locations where the gradients of the inverse mapping function are large. An example is given of a sheared X field, where a slow smooth continuous shear flow imposed on the boundary across one quasi-separatrix produces a flipping of magnetic field lines as they slip rapidly through the plasma in the other quasi-separatrix layer. It results in a strong plasma jetting localized in, and parallel to, the separatrix layers.

Nonlinear Force‐free Field Modeling of a Solar Active Region around the Time of a Major Flare and Coronal Mass Ejection

Solar flares and coronal mass ejections are associated with rapid changes in field connectivity and are powered by the partial dissipation of electrical currents in the solar atmosphere. A critical unanswered question is whether the currents involved are induced by the motion of preexisting atmospheric magnetic flux subject to surface plasma flows or whether these currents are associated with the emergence of flux from within the solar convective zone. We address this problem by applying state-of-the-art nonlinear force-free field (NLFFF) modeling to the highest resolution and quality vector-magnetographic data observed by the recently launched Hinode satellite on NOAA AR 10930 around the time of a powerful X3.4 flare. We compute 14 NLFFF models with four different codes and a variety of boundary conditions. We find that the model fields differ markedly in geometry, energy content, and force-freeness. We discuss the relative merits of these models in a general critique of present abilities to model the coronal magnetic field based on surface vector field measurements. For our application in particular, we find a fair agreement of the best-fit model field with the observed coronal configuration, and argue (1) that strong electrical currents emerge together with magnetic flux preceding the flare, (2) that these currents are carried in an ensemble of thin strands, (3) that the global pattern of these currents and of field lines are compatible with a large-scale twisted flux rope topology, and (4) that the ~1032 erg change in energy associated with the coronal electrical currents suffices to power the flare and its associated coronal mass ejection.

The effect of curvature on flux-rope models of coronal mass ejections

The large-scale curvature of a flux rope can help propel it outward from the Sun. Here we extend previous two-dimensional flux-rope models of coronal mass ejections to include the curvature force. To obtain analytical results, we assume axial symmetry and model the flux rope as a torus that encircles the Sun. Initially, the flux rope is suspended in the corona by a balance between magnetic tension, compression, and curvature forces, but this balance is lost if the photospheric sources of the coronal field slowly decay with time. The evolution of the system shows catastrophic behavior as occurred in previous models, but, unlike the previous models, flux ropes with large radii are more likely to erupt than ones with small radii. The maximum total magnetic energy that can be stored before equilibrium is lost is 1.53 times the energy of the potential field, and this value is less than the limiting value of 1.662 for the fully opened field. As a consequence, the loss of ideal MHD equilibrium that occurs in the model cannot completely open the magnetic field. However, the loss of equilibrium does lead to the sudden formation of a current sheet, and if rapid reconnection occurs in this sheet, then the flux rope can escape from the Sun. We also find that the held can gradually become opened without suffering any loss of equilibrium if the photospheric field strength falls below a critical value. This behavior is analogous to the opening of a spherically symmetric arcade in response to a finite amount of shear.

What is the source of the magnetic helicity shed by CMEs? The long-term helicity budget of AR 7978

An isolated active region (AR) was observed on the Sun during seven rotations, starting from its birth in July 1996 to its full dispersion in December 1996. We analyse the long-term budget of the AR relative magnetic helicity. Firstly, we calculate the helicity injected by differential rotation at the photospheric level using MDI/SoHO magnetograms. Secondly, we compute the coronal magnetic field and its helicity selecting the model which best fits the soft X-ray loops observed with SXT/Yohkoh. Finally, we identify all the coronal mass ejections (CMEs) that originated from the AR during its lifetime using LASCO and EIT/SoHO. Assuming a one to one correspondence between CMEs and magnetic clouds, we estimate the magnetic helicity which could be shed via CMEs. We find that differential rotation can neither provide the required magnetic helicity to the coronal field (at least a factor 2.5 to 4 larger), nor to the field ejected to the interplanetary space (a factor 4 to 20 larger), even in the case of this AR for which the total helicity injected by differential rotation is close to the maximum possible value. However, the total helicity ejected is equivalent to that of a twisted flux tube having the same magnetic flux as the studied AR and a number of turns in the interval [0.5, 2.0]. We suggest that the main source of helicity is the inherent twist of the magnetic flux tube forming the active region. This magnetic helicity is transferred to the corona either by the continuous emergence of the flux tube for several solar rotations (i.e. on a time scale much longer than the classical emergence phase), or by torsional Alfven waves.

MAGNETIC FIELD AND PLASMA SCALING LAWS: THEIR IMPLICATIONS FOR CORONAL HEATING MODELS

In order to test different models of coronal heating, we have investigated how the magnetic field strength of coronal flux tubes depends on the end-to-end length of the tube. Using photospheric magnetograms from both observed and idealized active regions, we computed potential, linear force-free, and magnetostatic extrapolation models. For each model, we then determined the average coronal field strength, B, in approximately 1000 individual flux tubes with regularly spaced footpoints. Scatter plots of B versus length, L, are characterized by a flat section for small L and a steeply declining section for large L. They are well described by a function of the form log = C1 + C2 log L + C3/2 log(L2 + S2), where C2 ≈ 0, -3 ≤ C3 ≤ -1, and 40 ≤ S ≤ 240 Mm is related to the characteristic size of the active region. There is a tendency for the magnitude of C3 to decrease as the magnetic complexity of the region increases. The average magnetic energy in a flux tube, B2, exhibits a similar behavior, with only C3 being significantly different. For flux tubes of intermediate length, 50 ≤ L ≤ 300 Mm, corresponding to the soft X-ray loops in a study by Klimchuk & Porter (1995), we find a universal scaling law of the form Lδ, where δ = -0.88 ± 0.3. By combining this with the Klimchuk & Porter result that the heating rate scales as L-2, we can test different models of coronal heating. We find that models involving the gradual stressing of the magnetic field, by slow footpoint motions, are in generally better agreement with the observational constraints than are wave heating models. We conclude, however, that the theoretical models must be more fully developed and the observational uncertainties must be reduced before any definitive statements about specific heating mechanisms can be made.

Current sheet formation in quasi-separatrix layers and hyperbolic flux tubes

In 3D magnetic field configurations, quasi-separatrix layers (QSLs) are defined as volumes in which field lines locally display strong gradients of connectivity. Considering QSLs both as the preferential locations for current sheet development and magnetic reconnection, in general, and as a natural model for solar flares and coronal heating, in particular, has been strongly debated issues over the past decade. In this paper, we perform zero-β resistive MHD simulations of the development of electric currents in smooth magnetic configurations which are, strictly speaking, bipolar though they are formed by four flux concentrations, and whose potential fields contain QSLs. The configurations are driven by smooth and large-scale sub-Alfvenic footpoint motions. Extended electric currents form naturally in the configurations, which evolve through a sequence of quasi non-linear force-free equilibria. Narrow current layers also develop. They spontaneously form at small scales all around the QSLs, whatever the footpoint motions are. For long enough motions, the strongest currents develop where the QSLs are the thinnest, namely at the Hyperbolic Flux Tube (HFT), which generalizes the concept of separator. These currents progressively take the shape of an elongated sheet, whose formation is associated with a gradual steepening of the magnetic field gradients over tens of Alfven times, due to the different motions applied to the field lines which pass on each side of the HFT. Our model then self-consistently accounts for the long-duration energy storage prior to a flare, followed by a switch-on of reconnection when the currents reach the dissipative scale at the HFT. In configurations whose potential fields contain broader QSLs, when the magnetic field gradients reach the dissipative scale, the currents at the HFT reach higher magnitudes. This implies that major solar flares which are not related to an early large-scale ideal instability, must occur in regions whose corresponding potential fields have broader QSLs. Our results lead us to conjecture that physically, current layers must always form on the scale of the QSLs. This implies that electric currents around QSLs may be gradually amplified in time only if the QSLs are broader than the dissipative length-scale. We also discuss the potential role of QSLs in coronal heating in bipolar configurations made of a continuous distribution of flux concentrations.

CRITERIA FOR FLUX ROPE ERUPTION: NON-EQUILIBRIUM VERSUS TORUS INSTABILITY

The coronal magnetic configuration of an active region typically evolves quietly for a few days before becoming suddenly eruptive and launching a coronal mass ejection (CME). The precise origin of the eruption is still under debate. The loss of equilibrium, or an ideal magnetohydrodynamic (MHD) instability such as torus instability are among the several mechanisms that have proposed to be responsible for the sudden eruptions. Distinct approaches have also been formulated for limited cases having circular or translation symmetry. We revisit the previous theoretical approaches setting them in the same analytical framework. The coronal field results from the contribution of a non-neutralized current channel added to a background magnetic field, which in our model is the potential field generated by two photospheric flux concentrations. The evolution on short Alfvenic timescale is governed by ideal MHD. We first show analytically that the loss of equilibrium and the stability analysis are two different views of the same physical mechanism. Second, we identify that the same physics is involved in the instabilities of circular and straight current channels. Indeed, they are just two particular limiting cases of more general current paths. A global instability of the magnetic configuration is present when the current channel is located at a coronal height, h, large enough so that the decay index of the potential field, ∂ln |B p|/∂ln h, is larger than a critical value. At the limit of very thin current channels, previous analysis has found critical decay indices of 1.5 and 1 for circular and straight current channels, respectively. However, with current channels being deformable and as thick as that expected in the corona, we show that this critical index has similar values for circular and straight current channels, and is typically in the range [1.1,1.3].

MAGNETIC ENERGY AND HELICITY FLUXES AT THE PHOTOSPHERIC LEVEL

The source of coronal magnetic energy and helicity lies below the surface of the Sun, probably in the convective zone dynamo. Measurements of magnetic and velocity fields can capture the fluxes of both magnetic energy and helicity crossing the photosphere. We point out the ambiguities which can occur when observations are used to compute these fluxes. In particular, we show that these fluxes should be computed only from the horizontal motions deduced by tracking the photospheric cut of magnetic flux tubes. These horizontal motions include the effect of both the emergence and the shearing motions whatever the magnetic configuration complexity is. We finally analyze the observational difficulties involved in deriving such fluxes, in particular the limitations of the correlation tracking methods.

A new model-independent method to compute magnetic helicity in magnetic clouds

Fil: Dasso, Sergio Ricardo. Consejo Nacional de Investigaciones Cientificas y Tecnicas. Oficina de Coordinacion Administrativa Ciudad Universitaria. Instituto de Astronomia y Fisica del Espacio. - Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Astronomia y Fisica del Espacio; Argentina